scholarly journals On the rate of convergence of interpolation polynomials of Hermite-Fejér type

1978 ◽  
Vol 19 (1) ◽  
pp. 29-37 ◽  
Author(s):  
J. Prasad
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Interpolation Polynomial ◽  
Image Position ◽  
Interpolation Polynomials

For the interpolation polynomial of Hermite-Fejér type An[f] of degree less than or equal to 4n − 1 constructed on the nodes , k = 1, 2, …, n, it is shown that for f ∈ CM(Ω) the inequalityholds where CM(Ω) is the class of continuous functions on [−1, 1] satisfying certain conditions, Ω is a certain modulus of continuity, and c3 and M are positive constants.

A Note on the Rate of Convergence of Hermite-Fejér Interpolation Polynomials*

1974 ◽  
Vol 17 (2) ◽  
pp. 299-301 ◽  
Author(s):  
R. B. Saxena
Keyword(s):  
Rate Of Convergence ◽  
Chebyshev Polynomial ◽  
Interpolation Polynomial ◽  
Image Position ◽  
Interpolation Polynomials

The Hermite-Fejér interpolation polynomial Hn[f] of degree ≤2n—1 is defined by(1)Where(2)are the zeroes of Chebyshev polynomial of first kind Tn(x)=cos n(arc cos x).

scholarly journals The Rate of Convergence of Lupasq-Analogue of the Bernstein Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Heping Wang ◽  
Yanbo Zhang
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Bernstein Operators ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous

We discuss the rate of convergence of the Lupasq-analogues of the Bernstein operatorsRn,q(f;x)which were given by Lupas in 1987. We obtain the estimates for the rate of convergence ofRn,q(f)by the modulus of continuity off, and show that the estimates are sharp in the sense of order for Lipschitz continuous functions.

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

scholarly journals Approximation properties of (p;q)-variant of Stancu-Schurer operators

2018 ◽  
Vol 37 (4) ◽  
pp. 137-151 ◽  
Author(s):  
Abdul Wafi ◽  
Nadeem Rao ◽  
_ Deepmala
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Continuous Functions ◽  
Second Order ◽  
Lipschitz Class ◽  
Approximation Properties ◽  
Direct Approximation ◽  
Korovkin Type Theorems

In this article, we have introduced (p;q)-variant of Stancu-Schurer operators and discussed the rate of convergence for continuous functions. We have also discussed recursive estimates, Korovkin-type theorems and direct approximation results using second order modulus of continuity, Peetre’s K-functional and Lipschitz class.

scholarly journals A link between Lebesgue constants and Hermite-Fejér interpolation

1986 ◽  
Vol 33 (2) ◽  
pp. 207-218 ◽  
Author(s):  
S. J. Goodenough
Keyword(s):  
Rate Of Convergence ◽  
Lagrange Interpolation ◽  
Historical Perspective ◽  
Lebesgue Constant ◽  
The Other ◽  
Lebesgue Constants ◽  
Close Link ◽  
The One ◽  
Image Position ◽  
Interpolation Polynomials

A review of the development of estimates for Lebesgue constants associated with Lagrange interpolation on the one hand, and estimates for the rate of convergence of Hermite-Fejér interpolation on the other hand, provides a historical perspective for the following surprising, close link between these apparently diverse concepts. Denoting by Λn (T) the Lebesgue constant of order n and by Δn (T) the maximum interpolation error for functions of class Lip 1 by Hexmite-Fejér interpolation polynomials of degree not exceeding 2n − 1, based on the zeros of the Chebyshev polynomial of first kind, we discover that, for even values of n, Λn(T) = n Δn(T).

scholarly journals On the Durrmeyer-Type Modification of Some Discrete Approximation Operators

1994 ◽  
Vol 1 (5) ◽  
pp. 523-536
Author(s):  
Paulina Pych-Taberska
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Approximation Theory ◽  
Continuous Functions ◽  
Discrete Approximation ◽  
Approximation Operators ◽  
Discrete Operators

Abstract In [Kratz and Stadtmüller, J. Approximation Theory 54: 326-337, 1988], for continuous functions f from the domain of certain discrete operators Ln the inequalities are proved concerning the modulus of continuity of Lnf. Here we present analogues of the results obtained for the Durrmeyer-type modification of Ln . Moreover, we give the estimates of the rate of convergence of in Hölder-type norms

The Analytic Character of the Birkhoff Interpolation Polynomials

1982 ◽  
Vol 34 (3) ◽  
pp. 765-768
Author(s):  
G. G. Lorentz
Keyword(s):  
Entire Function ◽  
Hermite Interpolation ◽  
Interpolation Polynomial ◽  
Birkhoff Interpolation ◽  
Interpolation Matrix ◽  
Image Position ◽  
Interpolation Polynomials

Let E be an m × (n + 1) regular interpolation matrix with elements ei, k = (E)i, k which are zero or one, with n + 1 ones. Then for each f ∈ Cn[a, b] and each set of knots X: a ≦ x1 < … < xm ≦ b, there is a unique interpolation polynomial P(f, E, X; t) of degree ≦ n which satisfies1A recent paper [1] discussed the continuity of P, as a function of x1, …,xm(with coalescences allowed). We would like to study in this note the analytic character of P as a function of real or complex knots X: x1, …, xm. This is easy for the Lagrange or the Hermite interpolation. In this case P is a polynomial in x1, …, xm if f is a polynomial, and an entire function in x1, …, xm if f is entire.

scholarly journals On the Convergence of a Family of Chlodowsky Type Bernstein-Stancu-Schurer Operators

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Lian-Ta Shu ◽  
Guorong Zhou ◽  
Qing-Bo Cai
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Weighted Approximation ◽  
Local Approximation ◽  
Continuous Functions ◽  
Lipschitz Continuous Functions ◽  
Lipschitz Continuous ◽  
Approximation Theorems ◽  
New Family ◽  
Weighted Modulus

We construct a new family of univariate Chlodowsky type Bernstein-Stancu-Schurer operators and bivariate tensor product form. We obtain the estimates of moments and central moments of these operators, obtain weighted approximation theorem, establish local approximation theorems by the usual and the second order modulus of continuity, estimate the rate of convergence, give a convergence theorem for the Lipschitz continuous functions, and also obtain a Voronovskaja-type asymptotic formula. For the bivariate case, we give the rate of convergence by using the weighted modulus of continuity. We also give some graphs and numerical examples to illustrate the convergent properties of these operators to certain functions and show that the new ones have a better approximation to functions f for one dimension.

Periodic solutions to functional differential equations

1985 ◽  
Vol 101 (3-4) ◽  
pp. 253-271 ◽  
Author(s):  
O. A. Arino ◽  
T. A. Burton ◽  
J. R. Haddock
Keyword(s):  
Differential Equations ◽  
Functional Differential Equations ◽  
Sufficient Conditions ◽  
Uniform Boundedness ◽  
Continuous Functions ◽  
Ultimate Boundedness ◽  
Space Of Continuous Functions ◽  
Uniform Ultimate Boundedness ◽  
Functional Differential ◽  
Image Position

SynopsisWe consider a system of functional differential equationswhere G: R × B → Rn is T periodic in t and B is a certain phase space of continuous functions that map (−∞, 0[ into Rn. The concepts of B-uniform boundedness and B-uniform ultimate boundedness are introduced, and sufficient conditions are given for the existence of a T-periodic solution to (1.1). Several examples are given to illustrate the main theorem.

The Dunkl generalization of Stancu type q-Szasz-Mirakjan-Kantorovich operators and some approximation results

2018 ◽  
Vol 34 (3) ◽  
pp. 363-370
Author(s):  
M. MURSALEEN ◽  
◽  
MOHD. AHASAN ◽  
Keyword(s):  
Rate Of Convergence ◽  
Modulus Of Continuity ◽  
Exponential Function ◽  
Second Order ◽  
Lipschitz Functions ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Approximation Properties ◽  
Kantorovich Operators ◽  
Korovkin’S Theorem

In this paper, a Dunkl type generalization of Stancu type q-Szasz-Mirakjan-Kantorovich positive linear operators ´ of the exponential function is introduced. With the help of well-known Korovkin’s theorem, some approximation properties and also the rate of convergence for these operators in terms of the classical and second-order modulus of continuity, Peetre’s K-functional and Lipschitz functions are investigated.

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